The chapters of this Handbook volume covers nine main topics that are representative of recent
theoretical and algorithmic developments in the field. In addition to
the nine papers that present the state of the art, there is an article on
the early history of the field.

The handbook will be a useful reference to experts in the field
as well as students and others who want to learn about discrete
optimization.

All of the chapters in this handbook are written by authors who have
made significant original contributions to their topics. Herewith a brief introduction to the chapters
of the handbook.

"On the history of combinatorial optimization (until 1960)" goes back to work of Monge in the 18th century on the assignment problem and presents six problem areas: assignment, transportation,
maximum flow, shortest tree, shortest path and traveling salesman.

The branch-and-cut algorithm of integer programming is the
computational workhorse of discrete optimization. It provides the
tools that have been implemented in commercial software such as CPLEX
and Xpress MP that make it possible to solve practical problems in
supply chain, manufacturing, telecommunications and many other areas.
"Computational integer programming and cutting planes" presents the key ingredients
of these algorithms.

Although branch-and-cut based on linear programming relaxation is the
most widely used integer programming algorithm, other approaches are
needed to solve instances for which branch-and-cut performs poorly
and to understand better the structure of integral polyhedra. The
next three chapters discuss alternative approaches.

"The structure of group relaxations" studies a family of polyhedra obtained by dropping certain
nonnegativity restrictions on integer programming problems.

Although integer programming is NP-hard in general, it is
polynomially solvable in fixed dimension. "Integer programming, lattices, and results in fixed dimension" presents results in this area including algorithms that use reduced bases of integer lattices that are capable of solving certain classes of integer programs that defy
solution by branch-and-cut.

Relaxation or dual methods, such as cutting plane algorithms,
progressively remove infeasibility while maintaining optimality to
the relaxed problem. Such algorithms have the disadvantage of
possibly obtaining feasibility only when the algorithm terminates.
Primal methods for integer programs, which move from a feasible
solution to a better feasible solution, were studied in the 1960's
but did not appear to be competitive with dual methods. However,
recent development in primal methods presented in "Primal
integer programming" indicate that this approach is not just interesting theoretically but may have
practical implications as well.

The study of matrices that yield integral polyhedra has a long
tradition in integer programming. A major breakthrough occurred in
the 1990's with the development of polyhedral and structural results
and recognition algorithms for balanced matrices. "Balanced matrices" is a tutorial on the
subject.

Submodular function minimization generalizes some linear
combinatorial optimization problems such as minimum cut and is one of
the fundamental problems of the fieldthat is solvable in polynomial
time. "Submodular function minimization" presents the theory and algorithms of this subject.

In the search for tighter relaxations of combinatorial optimization
problems, semidefinite programming provides a generalization of
linear programming that can give better approximations and is still
polynomially solvable. This subject is discussed in "Semidefinite programming and integer programming."

Many real world problems have uncertain data that is known only
probabilistically. Stochastic programming treats this topic, but
until recently it was limited, for computational reasons, to
stochastic linear programs. Stochastic integer programming is now a
high profile research area and recent developments are presented in
"Algorithms for stochastic mixed-integer programming
models."

Resource constrained scheduling is an example of a class of
combinatorial optimization problems that is not naturally formulated
with linear constraints so that linear programming based methods do
not work well. "Constraint programming" presents an alternative enumerative approach
that is complementary to branch-and-cut. Constraint programming,
primarily designed for feasibility problems, does not use a
relaxation to obtain bounds. Instead nodes of the search tree are
pruned by constraint propagation, which tightens bounds on variables
until their values are fixed or their domains are shown to be empty.

• ISBN: 0444515070
• ISBN-13: 9780444515070
• Editors : K. Aardal and G. L. Nemhauser and R. Weismantel
• Category : Business & Economics - Management Science
• Language : English
• Format:  Hardcover
• Publication Date: 10/1/2005
• Publisher Imprint: North-Holland
• Price ($AUD): $416.95


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